Rod on a cylinder

When the beetle reaches the end of the rod, let the rod rotates through an angle $α$ from it's initial orientation. Then the normal reaction of the cylinder on the rod is

$N=(m+0.2m)g\cos α=1.2mg\cos α$ .

Moment balance equation about the point of contact of the rod and the cylinder gives

$mgrα\cos α=0.2mg(πr-rα)\cos α\Rightarrow α=\dfrac{π\times 0.2m}{m+0.2m}=\dfrac{π}{6}$ .

For static equilibrium of the rod over the cylinder, the friction force $F_f$ is given by $F_f=(m+0.2m)g\sin α$ . This must be less than or equal to it's limiting value $\mu_sN=\mu_s(m+0.2m)g\cos α$ , where $\mu_s$ is the coefficient of static friction. This implies

$\mu_s\geq \tan α$ .

Substituting the value of $α$ we get $\mu_s\geq \tan \dfrac{π}{6}=\dfrac{1}{√3}\approx \boxed {0.5773}$